Abstract This paper deals with the problem of asymptotic breeding/burning waves during long term nuclear fission processes. The uranium–plutonium (U–Pu) conversion cycle is considered under fast spectrum conditions. A one-group diffusion equation coupled with burn-up equations is set up. The nuclide atom number densities can be determined as functions of the neutron fluence only, as the natural radioactive processes are neglected. It is found then that the diffusion equation with the neutron fluence dependent macroscopic cross sections is analytically integrable (solvable) in the 1-D case without feedback effects. A permanent solitary wave solution exists under certain conditions, where the infinite medium multiplication factor first increases from a subcritical level up to a supercritical point and then falls to another subcritical level again, along with the increasing neutron fluence. Relationships between wave amplitude, wave number, parameters of fuel are studied. A representative example is shown for a breeding/burning solitary wave propagating in a 238 U medium with a suitable content of burnable poison, where the conversion chain is considered up to 242 Pu. Finally it is demonstrated as well that in a two-dimensional cylindrical case, a multi-dimensional permanent solitary wave pattern with a constant drift speed can be also achieved based on the 1-D fundamental solution by adjusting the initial radial distribution of the fuel composition, where more fuel and higher enrichment are needed in the outer peripheral region than in the inner one.